How can I display an entire page at a specific path/url in a block?
If you do something like this...

D6:

$foo=menu_execute_active_handler('path/to/the/homebox/page');


D7:

$foo=menu_execute_active_handler('path/to/the/homebox/page', FALSE);


...$foo should include either all of the html or a renderable array of the path/page content in question so in theory you could then stuff that in a block, either via a custom module or (ick) using the standard block interface and the (double ick) php input filter.

MORE THOUGHTS:

My basic thought above was to try to get to the menu handler for the page/path in question with as little custom code as possible.  But, if you look in the HomeBox module, all it is doing for each page/path is calling homebox_build() along with a page argument that is easily crafted up.  So, if you are comfortable with coding, you could write a custom module that uses that function within its block hooks to create the content.

Also, not sure about any css/js that would be needed but that could be hand crafted I suppose if necessary.