I will expand on Bacon's comment.
When you are married, and you acquire any kind of property, you automatically get a legal agreement. In most states that property is owned jointly and while there are exceptions that is the case most of the time.
When you are unmarried, there is no such assumption of joint acquisition. While words might be said differently between the two parties, if there is nothing written down and signed then courts will almost always assume that only one party owns the property.
Now unmarried people go into business all the time, but they do so by creating legally binding agreements that cover contingencies. If you two do proceed with this plan, it is necessary to create those documents with the help of a lawyer. Although expensive paying for this protection is a small price in relation to what will probably be one of the largest purchases in your lives.
However, I do not recommend this. If Clayton can and wants to buy a home he should. Emma can rent from Clayton. That rent could any amount the two agree on, including zero. If the two do get married, well then Emma will end up owning any equity after that date. If they stay together until death, it is likely that she (or her heirs) will own half of it anyway. Also if this house is sold, the equity pass into larger house they buy after marriage, then that will be owned jointly.
If they do break up, the break up is clean and neat. Presumably she would have paid rent anyway, so nothing is lost. Many people run into trouble having to sell at a bad time in a relationship that coincides with a weak housing market. In that case, both parties lose.
So much like Bacon's advice I would not buy jointly. There is no upside, and you avoid a lot of downside.
Don't play "house" by buying a home jointly when you are unmarried.