Will server's php ver 5.5 work with Boonex 7.2?

We're currently using Boonex 7.2. Our host provider wants us to upgrade our VPS to php 5.5. Will that effect our site if we do?

You should not see any issues but you need to question why your host wants you to upgrade; it is a VPS, there should be no reason for host to force you to upgrade against your wishes.  Have you asked them why they want you to upgrade?

php 5.4 EOL was 5 months ago, and assuming its managed VPS, they are doing the right thing.

Yes, that is true but that doesn't automatically translate into not using it.  In fact the Dolphin official documentation reads:

PHP 5.3.0 or higher (or at least PHP 5.2.0 for Dolphin 7.1.x)

Of course if new security issues are found, they are not likely to be fixed because it is end of life.  It it was me and I had a working Dolphin on the VPS I would want a clear reason why the version should be upgraded; otherwise, I would say no, leave my VPS alone.

I will say this, before you allow them to upgrade the PHP version on the VPS, know the exact version they are going to upgrade to so you can ask if anyone has any issues with that PHP version with Dolphin.

Here is why I would be hesitant, what if it takes your site down?  Can you afford to have your site go down for some period of time?  What plans does your host have in case the site goes down, will they image the VPS so you don't lose anything?  Ask questions before just saying, "Go ahead."