Q:

Can I make the subsection CounterBox continue counting across section boundaries?

This is what I used in a stylesheet for Section 
CellDingbat->Cell[TextData[{CounterBox["Section"], "."}]]

This is what I used in a stylesheet for Subsection 
CellDingbat->Cell[TextData[{CounterBox["Subsection"], "."}]]

As you can see, the subsection counter get reset when I start a new section. How can I get the subsection counter to continue its count in the new section.

A:

The thing which resets the "Subsection" counter is the CounterAssignments option.  The option includes a list of assignments, including:
{"Subsection", 0}

which just means that, beginning in that cell, the "Subsection" counter is assigned to 0 (and will be incremented to 1 beginning at the next cell).
You can use the Option Inspector to change the option on a specific Section cell, or you can change the stylesheet for the entire notebook.  To change the stylesheet, choose the Format->Edit Stylesheet... menu item, then paste and interpret the following cell expression at the end of the stylesheet:
Cell[StyleData["Section"], CounterAssignments->
    {{"Subsubsection", 0}, {"Item", 0}, {"Subitem", 0}, {"Subsubitem", 0},
     {"ItemNumbered", 0}, {"SubitemNumbered", 0}, {"SubsubitemNumbered", 0}}]

This just mimics the default list for the Section style, but omitting the "Subsection" counter setting.

A:

By default (I mean in Default.nb style sheet) the Section style resets the subsection counter and many others, by means of the following option:
CounterAssignments->{{"Subsection", 0},{"Subsubsection", 0}, {"Item", 0}, {"Subitem", 0}, {
   "Subsubitem", 0}, {"ItemNumbered", 0}, {"SubitemNumbered", 0}, {"SubsubitemNumbered", 0}}

So, if you want some of such headings not to re-start the counter after a section, just do the following:
Open the notebook's style sheet and add a Section style and then use the option 
CounterAssignments->{{"Subsection", 0},{"Subsubsection", 0}, {"Item", 0}, {"Subitem", 0}, {
   "Subsubitem", 0}, {"ItemNumbered", 0}, {"SubitemNumbered", 0}, {"SubsubitemNumbered", 0}}

adding or removing the counters you want to modify (add a counter to reset after each Section, remove from the list to not reset).
In our case, you should remove the {"Subsection", 0} from the list.
Here is a picture of what you should get.

