user-1: With objectless mindfulness, if I am in a bad mood the mood perseveres. I just observe the bad mood but it doesn't easily stop. 
However, if I applied concentration on an object such as the breath, the mood immediately dissipates and even sometimes vanishes.
Does strong concentration repress emotions, or does it process them in a healthy way? In other words, does focusing on the breath still permit the emotions to be experienced naturally despite an absence of thought or feeling?

user8157: There is nothing unhealthy about annihilating unwholesome emotions. 

As I noticed that it leads to my own affliction, it subsided. As I noticed that it leads to the affliction of others... to the affliction
  of both... it obstructs discernment, promotes vexation & does not lead
  to Unbinding, it subsided. Whenever thinking imbued with sensuality
  had arisen, I simply abandoned it, dispelled it, wiped it out of
  existence.
MN 19


user14060: Well you do not say what experience is this ''objectless mindfulness'', which is an odd phrasing, because, beyond the bad word that is mindfulness, there is always an object of concentration or contemplation.
And the whole point of contemplation is either to know  or track the kaya, the mano and the citta as well as to calm the kaya, the mano and the citta:

it is easy to track the kaya, mano, citta, for any puthujjana who has sati and is good at tranquilising the kaya, mano and citta
it is easy to calm the kaya, mano and citta,  for any puthujjana who has sati and is good at tracking the kaya, mano, citta, thought, sanna, vedana (in one word, that's call sati samprajanya which is the method to stop the spreading of infatuation and hate towards whatever is loved and hated by puthujjanas)
in other word, you cannot be good at tranquilizing the kaya, mano and citta without tracking the mano, kaya and citta and you cannot be good at tracking the kaya, mano and citta, without being good at tranquilizing the kaya, mano and citta (which is done by seeing how nice it is to be calm and how awful it is to be agitated).

Fortunately, you claim to calm your mood with the breath, so that's good. Do not forget that after you become tranquil with the breath, you need to 
contemplate the impermanence of typically the 5 aggregates
SInce you like to contemplate things other than the breath, there are plenty of topics to contemplate here
https://www.accesstoinsight.org/tipitaka/an/an10/an10.060.piya.html
When you want to get the citta into samadhi, you have 4 famous samadhis here. the 2 samadhis with the clearest directions and clear exposition of results are:

And what is the development of concentration that, when developed & pursued, leads to the attainment of knowledge & vision? There is the case where a monk attends to the perception of light and is resolved on the perception of daytime [at any hour of the day]. Day [for him] is the same as night, night is the same as day. By means of an awareness open & unhampered, he develops a brightened mind. This is the development of concentration that, when developed & pursued, leads to the attainment of knowledge & vision. 
And what is the development of concentration that, when developed & pursued, leads to mindfulness & alertness? There is the case where feelings are known to the monk as they arise, known as they persist, known as they subside. Perceptions are known to him as they arise, known as they persist, known as they subside. Thoughts are known to him as they arise, known as they persist, known as they subside. This is the development of concentration that, when developed & pursued, leads to mindfulness & alertness. 

www.buddha-vacana.org/sutta/anguttara/04/an04-041.html
If the last one is what you call ''objectless mindfulness'', then call it properly by calling it what it is : training for sati samprajanya instead of inventing new words that nobody knows what they mean.

user-1: 
With objectless mindfulness, if I am in a bad mood the mood
  perseveres. I just observe the bad mood but it doesn't easily stop.

This may be because the attention you're giving to the bad mood is the wrong kind of attention. There may be a clue in you wanting for the bad mood to easily stop. Observe with no anticipation for anything.

However, if I applied concentration on an object such as the breath,
  the mood immediately dissipates and even sometimes vanishes.

The dissipation of the mood by shifting focus may be a further clue to the mental proliferation that you could be adding to the 'watching' of this bad mood. 

Does strong concentration repress emotions, or does it process them in
  a healthy way? In other words, does focusing on the breath still
  permit the emotions to be experienced naturally despite an absence of
  thought or feeling?

It doesn't need to be strong concentration. One just needs to simply switch focus to something else. The something else can become a further problem like an escape though. Notice breath, movement, objects, light gradients... There are many wonderful harmless things to remove us from mental anguish. This is, indeed, healthy and not a repression technique. 

-1: <p>With objectless mindfulness, if I am in a bad mood the mood perseveres. I just observe the bad mood but it doesn't easily stop. </p>

<p>However, if I applied concentration on an object such as the breath, the mood immediately dissipates and even sometimes vanishes.</p>

<p><strong>Does strong concentration <em>repress</em> emotions, or does it process them in a healthy way? In other words, does focusing on the breath still permit the emotions to be experienced naturally despite an absence of thought or feeling?</strong></p>
 8157: <p>There is nothing unhealthy about annihilating unwholesome emotions. </p>

<blockquote>
  <p><em>As I noticed that it leads to my own affliction, it subsided. As I noticed that it leads to the affliction of others... to the affliction
  of both... it obstructs discernment, promotes vexation &amp; does not lead
  to Unbinding, it subsided. Whenever thinking imbued with sensuality
  had arisen, I simply abandoned it, dispelled it, <strong>wiped it out of
  existence</strong>.</em></p>
  
  <p><em>MN 19</em></p>
</blockquote>
 14060: <p>Well you do not say what experience is this ''objectless mindfulness'', which is an odd phrasing, because, beyond the bad word that is mindfulness, there is always an object of concentration or contemplation.
And the whole point of contemplation is either to know  or track the kaya, the mano and the citta as well as to calm the kaya, the mano and the citta:</p>

<ul>
<li><p>it is easy to track the kaya, mano, citta, for any puthujjana who has sati and is good at tranquilising the kaya, mano and citta</p></li>
<li><p>it is easy to calm the kaya, mano and citta,  for any puthujjana who has sati and is good at tracking the kaya, mano, citta, thought, sanna, vedana (in one word, that's call sati samprajanya which is the method to stop the spreading of infatuation and hate towards whatever is loved and hated by puthujjanas)</p></li>
<li><p>in other word, you cannot be good at tranquilizing the kaya, mano and citta without tracking the mano, kaya and citta and you cannot be good at tracking the kaya, mano and citta, without being good at tranquilizing the kaya, mano and citta (which is done by seeing how nice it is to be calm and how awful it is to be agitated).</p></li>
</ul>

<p>Fortunately, you claim to calm your mood with the breath, so that's good. Do not forget that after you become tranquil with the breath, you need to 
contemplate the impermanence of typically the 5 aggregates</p>

<p>SInce you like to contemplate things other than the breath, there are plenty of topics to contemplate here</p>

<p><a href="https://www.accesstoinsight.org/tipitaka/an/an10/an10.060.piya.html" rel="nofollow noreferrer">https://www.accesstoinsight.org/tipitaka/an/an10/an10.060.piya.html</a></p>

<p>When you want to get the citta into samadhi, you have 4 famous samadhis here. the 2 samadhis with the clearest directions and clear exposition of results are:</p>

<blockquote>
  <p>And what is the development of concentration that, when developed &amp; pursued, leads to the attainment of knowledge &amp; vision? There is the case where a monk attends to the perception of light and is resolved on the perception of daytime [at any hour of the day]. Day [for him] is the same as night, night is the same as day. By means of an awareness open &amp; unhampered, he develops a brightened mind. This is the development of concentration that, when developed &amp; pursued, leads to the attainment of knowledge &amp; vision. </p>
  
  <p>And what is the development of concentration that, when developed &amp; pursued, leads to mindfulness &amp; alertness? There is the case where feelings are known to the monk as they arise, known as they persist, known as they subside. Perceptions are known to him as they arise, known as they persist, known as they subside. Thoughts are known to him as they arise, known as they persist, known as they subside. This is the development of concentration that, when developed &amp; pursued, leads to mindfulness &amp; alertness. </p>
</blockquote>

<p>www.buddha-vacana.org/sutta/anguttara/04/an04-041.html</p>

<p>If the last one is what you call ''objectless mindfulness'', then call it properly by calling it what it is : training for sati samprajanya instead of inventing new words that nobody knows what they mean.</p>
 -1: <blockquote>
  <p>With objectless mindfulness, if I am in a bad mood the mood
  perseveres. I just observe the bad mood but it doesn't easily stop.</p>
</blockquote>

<p>This may be because the attention you're giving to the bad mood is the wrong kind of attention. There may be a clue in you wanting for the bad mood to easily stop. Observe with no anticipation for anything.</p>

<blockquote>
  <p>However, if I applied concentration on an object such as the breath,
  the mood immediately dissipates and even sometimes vanishes.</p>
</blockquote>

<p>The dissipation of the mood by shifting focus may be a further clue to the mental proliferation that you could be adding to the 'watching' of this bad mood. </p>

<blockquote>
  <p>Does strong concentration repress emotions, or does it process them in
  a healthy way? In other words, does focusing on the breath still
  permit the emotions to be experienced naturally despite an absence of
  thought or feeling?</p>
</blockquote>

<p>It doesn't need to be strong concentration. One just needs to simply switch focus to something else. The something else can become a further problem like an escape though. Notice breath, movement, objects, light gradients... There are many wonderful harmless things to remove us from mental anguish. This is, indeed, healthy and not a repression technique. </p>
