There are many features which can only be accomplished in Acrobat Pro with form fields that can't be done in InDesign.
Putting a background color can be done in Acrobat. you'll have to open the PDF in Acrobat. Right-click on a checkbox (or other field) and choose Properties > Appearance.
Acrobat doesn't use styles for form fields, but if you can multiple-select checkboxes (or other form fields) and apply properties on all of them at the same time in the Properties dialog.
Thanks for the reply. I don't see how I can change the background color of the ticked box. Only the default color. Am I missing something?
Checkbox form fields can only have one background. They don't have states.
Buttons can have different backgrounds for different states.
That is just the way PDF forms work.