Can I use a Sharkbite connector to join copper to rubber pipe?
SharkBite® only officially supports 4 types of pipe, according to the FAQ

Q:  What types of pipe can SharkBite fittings be used with?
A:  SharkBite fittings are certified for use with:

Copper pipe hard drawn Type K, L and M and annealed Type M not to exceed 3/8 nominal, complying with ASTM B88
PEX pipe complying with ASTM F 876 or CSA B137.5
CPVC pipe complying with ASTM D 2846 or CSA B137.6
PE-RT pipe complying with ASTM F 276


If you're trying to use the fittings with other types of pipe or tube, there's no guarantee that it will work.

If you're plumbing a drinking fountain, you'll want to make sure you use materials listed and labeled for use with potable water. &quot;Rubber hose&quot; is not a technical term, so there's no way to tell what type of pipe it actually is. So before you continue this project, you'll want to figure out exactly what type of pipe you're using, and make sure it's safe to use with drinking water.
Since this fixture is going to be outside, you also might want to consider installing a check valve somewhere on the supply plumbing. This is done to protect the rest of the water supply, to help reduce the risk of contamination.