One objection I keep hearing is "your tax dollars already pay for abortion, this is no different." It is. First off, according to president Obama no tax dollars pay for abortions, so right of the bat you are calling your president a liar. But truth be told, he is a liar, because on day 1 of his presidency in 2009 he abolished the Mexico City Policy, which forbade US tax dollars funding abortions in foreign lands.
But paying for these things remotely via tax dollars is very different from paying for them directly, and I think an analogy can illustrate why.
Let's say I buy groceries at the local grocery store. Now, the owner of that store is a psychopath, and takes the money from the register after I leave, buys ammo, and goes on a shooting spree at a nearby school. I might feel shocked, but I am not morally nor legally responsible.
Now let's say instead, the grocery store owner tells me he needs money to buy ammo to go on a shooting spree, and I hand him a $20 saying "this one's on me." I will be doing time as an accessory to murder. The law is pretty clear on when I am responsible for the evil committed and when I am not.
And so no - paying taxes, even if some of that tax money will be used against my wishes to pay for abortions, is very different in terms of moral responsibility from specifically paying for abortion through the HHS mandate. There is a fundamental difference in the degree of cooperation with evil.