I'm with Red. Bitter, it's just not that simple for couples who come together later with children from previous relationships, with one owning a house and the other not, etc.
I own my house. DP pays half of all utility bills, TV, phone, council tax (which we'd each expect to do, whoever owned the house). He pays half the interest part of the mortgage (not capital, which I pay because it's my asset). This and his bills share equate to a set amount, and we have a lodger agreement in place covering this. This works for us - we're both happy and think it's fair. I keep my asset. He has the freedom to put some money away if he has it/wants to.
What's different in your scenario is that you sound like you're starting out, pre-children, and you may have his child, and this may mean being on reduced income for a time (maternity leave/working reduced hours). And so while I think it's equitable to pay half of house-related costs when both earning reasonably well, if you're off work with his baby, he should be paying your share.
If you were renting, you'd have to pay council tax, water rates, etc, whilst not owning the property. I don't see why you feel you should have a claim to the property if you're paying these (whilst not on a mortgage)?
There's some good advice up-thread about marriage, trusts, mortgages with your work contract, etc, which I don't know about (not relevant to our circumstances). For if I die, we have good protection in place to ensure DP and all the kids can continue living in the house, while retaining the asset for my children.