The individual mandate used to be a conservative idea. As discussed in this piece which appeared in the Wall Street Journal, the mandate was originally pushed by conservative leaders like the Heritage Foundation, Mitt Romney and Newt Gingrich. Richard Nixon supported an employer mandate, an idea to the left of Obama's plan. Even Milton fuckin' Friedman, a very conservative (and, to be fair, a very insightful and influential) economist, advocated for "a requirement that every U.S. family unit have a major medical insurance policy."
Most opposition to the mandate centers around the distinction between regulating economic activity and forcing economic activity. It is one thing, they say, for government to put rules in place regarding activity that you would want to do anyway (e.g., you want to buy a car, and the government can put rules in place about safety, pollution, etc.). But it's another thing, they say, to force you to participate in an activity you wouldn't otherwise choose to be involved in. And you know what? They kinda have a point.
But here's the thing: whether ObamaCare is repealed or not, simple biology mandates that everyone is involved in the health care industry. At some point, we are going to get sick and need medical care. Given that we will all eventually need care, and the costs to the system involved in such care, society has a role, via government, in regulating how that care is paid for.