Radical equation - can I square both sides with more than 1 radical on one side? I'm familiar with equations like:

$\sqrt{x+1} - \sqrt{x+2} = 0 $

Has no solutions, it's just an example off the top of my head

Just move the negative square root to the other side, square both sides and solve. 

$\sqrt{x+1} = \sqrt{x+2}$

$x+1 = x+2$

0 = 1

My question is, if there are two square roots on one side, then can I still square both sides in this way:

$\sqrt{x+1} - \sqrt{x+2} = \sqrt{x+3}$

$x+1 - (x+2) = x+3$

$x+1 - (x-2) = x+3$

$x = -4$

Or does squaring both sides cause something strange to happen on the left hand side?
No, what you did was wrong. 
this is because : 

$\sqrt{x+1} - \sqrt{x+2} = \sqrt{x+3}$ does not imply $(x+1)-(x+2)=x+3$.

You need to square both sides, which means:

$(\sqrt{x+1} - \sqrt{x+2})^2=x+3$

$(x+1)+(x+2)-2\sqrt{(x+1)(x+2)}=x+3$.

As a sidenote, you can plug x=-4 into the original equation to check whether x=-4 is not a solution.
How I explain it to myself is negative if you want to multiply both sides of the equation by the same number, say 3, you can multiply-across through the pluses and minuses by 3, or take it in stages by putting a bracket around both sides, but the 3 at the start of the brackets on both sides and then multiply across in through the brackets(effectively inflating both sides by the same amount)... but you can’t square across, you need to square both sides, and I find using Brackets here are always a big help.
And if you’ve got say, two square roots (radicals) added together (or subtracted) on one side then it’s an easier calculation to slide one over to the other side, change the plus to a minus , and vice versa, and then square both sides eg $\sqrt{2x-1} + \sqrt{x-1} = 5$   next step $\sqrt{2x-1} = 5 – \sqrt{x-1}$