Funny that you would ask this, I have a cold rn and got a passing thought about the fact that Gage probably wouldn't think/be able to blow his nose so Hunter would need to help him like we do kids ha ha
I think Hunter first move if Gage was clearly sick would be to put him to bed with maybe an ice pack and some water close and go to his business, but you're right that Gage would indeed seek Hunter if he was sick. Once he'd realize through the fever that he is alone he would crawl out of bed and look for him. Hunter would probably grumble and complain about it to make it clear that it annoys him while still wrapping him into a blanket and helping him blow his nose while letting him rest his head on his lap lolol so he would kind of openly care for him but he'd make sure that everyone else knows how annoying it is for him (〃￣ω￣〃ゞ what a big tsundere, making oats and broth with pasta just for Gage while complaining that he can't do his damn job with this stupid mutt rubbing his snot all over his pants (don't pay attention to the relief in his sigh once Gage's temperature finally starts dropping shhh sshhhh)