I know social pressure is tough, but fight back, be honest. This is mostly aimed at healers, but can be applied to anyone, just substitute the appropriate words to match your own situation.
How often are you offline because you don't want to heal? That's hiding. Maybe it's on an alt, maybe it's in another game, maybe it's in that strange place called Real Life. This doesn't apply to actually being called away. But why do you have to hide? Imagine a glorious world in which you can just say that you don't feel like healing.
Oh, people won't be happy. They're used to healers either healing when asked, or not being there at all. When you hide, you also hide your refusal, and therefore your choice.
So log in and say it: you don't feel like healing. At first it will be a shock. But give it time and someday, people won't feel entitled to your heals and you won't feel like hiding when you just want to do something else.
The rest of this is aimed at the people who are outraged. Except the part after that.
People don't always stay offline because they want to. Sometimes it's because logging in means being bombarded by urgent requests by you. Whether online or offline, that is their choice, to say yes or no. It's pointless to drive them offline, just because you think being online makes them your servant. That only encourages burnout, making it harder to find them in the future.
So when you see your healer friend on his alt and ask him to heal, take no for an answer, it's his right.
As you might have guessed, this was triggered by RealID and the ability to see friends on alts. It's coming. People will have to choose: do they want to refuse a friend invite or do they want to turn down requests for runs? I'm leaning toward the first, because while I regard it as hiding, it's also much more private, thanks to information chaining out to friends of friends. I already have my friends and we know our alts and can turn down whatever we don't want. We don't need to be tied into a friend friend with whom we don't share that understanding.