If it is ever possible to generalize about the happiness of any group of people, it would seem that Conservatives may have a belief system that could be associated with greater happiness.
Conservatives are people who have faith in the power and capabilities of individuals. When people believe in themselves and in others, the world in which they live feels like a more positive place.
They're also people who look at what the nation is, recognize that there are flaws in how things are done, but generally believe that the nation is what it is as a result of a system that works. When people have a positive view of their nation, believe that "the system works" (or at least that it has worked since the time of nations' birth), and value the principles on which the nation was founded; they tend to feel more satisfied with, and proud of, their nation.
Conservatives remain very committed to those principles on which the nation was founded, to the point where many almost feel as if they have a responsibility to uphold, and preserve for the future, those principles. Whenever people feel connected to their history in that way they can feel they have a little more sense of purpose or connection to their country.
Conservatives see one of the biggest challenges as that of trying to limit the size of government, as well as its involvement and limitations on individual lives. The challenge of trying to limit a creeping, increasingly oppressive, over-sized government; is a formidable one. Still, with its base in a general belief in the people, this challenge is more likely to stir up awareness of strength and independence - rather than hopelessness and helplessness. A belief in stemming the creeping influence/involvement of government in the lives of the people can come from an awareness of the ways in which government actually destroys lives; but, again, this awareness doesn't come from a belief that people cannot manage without large government. It is, quite simply, less depressing to have faith in fellow human beings.
While this is, of course, a generalized picture of views, Liberals focus on what has not worked and on their belief that individuals cannot thrive without government's help. Liberals focus on the very real problems of the "have not's" and believe they cannot "have" without government intervention in some way. Those Liberals who are, themselves, "have not's", have seen that the often over-simplified Conservative concept of "just doing what one one needs to do to get out" doesn't always work for some people. These can be people who have tried to do all the right things, to no avail. Liberals who, themselves, can be counted among the affluent, often have the awareness that not everyone gets the "breaks" they got. Of course, too, wealthy Liberals can have a sense of responsibility to the less fortunate, but belief that only the government can help the less fortunate generally contributes to an acute awareness that "all is not fair" in this world (or nation) (which it most definitely is not). An aim to make things fair, however, can cause the people committed to that aim to be acutely aware of what is wrong in the nation, rather than what is right.
For Liberals, though, it isn't just a matter of trying to make things fair. Liberals focus on any number of problems in the country/world and believe government must eradicate them. Liberals want laws about bike helmets, trans fats, smoking, seatbelts, and cell phones in cars. Unlike Conservatives who, as part of their valuing the individual, see "diversity" as being about the mind and soul of the individual; Liberals favor laws establishing diversity, based on gender, race, and sexual orientation. People who see and respect others as individuals tend to be happier than those who see others only as a gender, race, or sexual identity. People who believe that everyone else needs government to help them in their lives are people who either feel superior to others (in which case they live in a world where few others can be admired as capable); or else people who count themselves among the less capable (in which case they are less likely to feel satisfied and happy in life).
None of this is to say that all Conservatives are happy with things just the way they are. Most Conservatives don't live in their own little world of happiness, oblivious to the struggles of the less fortunate. In fact, these days more and more of the less fortunate (or at least less wealthy) may be turning to Conservative thinking, now that Liberalism has morphed into something very different from what it was forty years ago. Just as Conservatives tend to suspect, it turns out that the non-wealthy (and even the low-income) Americans are aware of, and value, the promise of the individual and the principles on which this nation was founded.
Neither is it to say, however, that all Liberals are miserably unhappy, low-income people; or else "bleeding heart", guilty-ridden, wealthy people who never get to enjoy life.
It is simply to say that, whether Conservative or Liberal, all people are individuals. It's also to say, though, that people who see, respect, and value others as the capable individuals they are (or have the potential to be if government doesn't get in the way) are most likely, and generally, happier.