The Historical Statistics of the United States: Colonial Times to 1970 (found here in pdf format from this website ) should hold accurate information.

Series A 6-8 on page 8 gives a total population of 180,671,000 (and a quick google check at least brings up the same number, for whatever that's worth).

Series H 800-805 on on page 391 gives a Catholic membership of 40,871,000. That would be 22.62%.

I think I have those numbers right. Make sure to take a look at the methodology statements on 389 and 390 if you're this is for a scholarly purpose. It mentions the differences in how Catholics came up with their membership numbers compared to some other denominations.

T H White's book "The Making of a President 1960" has a whole section on the demographics and impact on the campaign which i highly recommend.