Only in California and Arizona do you have any assumed rights. Beyond that, yes she can as you forfeited your assume claim by not being married to her at the time she got pregnant. Now, you have to petition the court to request any rights. If she's been gone less than six months, you can file there, otherwise it will be in her state, so you will need to find her. Also, at any time, before the child turns 18, she can file for retroactive child support on you. As such, you need to contact child support enforcement and determine how much child support you should be paying. Start putting that money into a trust in the child's name.
My nursing home roommate got hit with $125,000 in retroactive child support after he finally found his son, which pissed off the mother. She's not getting much of it, as he died of cancer two weeks after the boy came to see him. Now there's a lien on his estate, so his younger children will be getting nothing.
You have an up mountain, not hill, battle ahead of you. In a word, prepare, prepare, prepare.
When you locate her, get an order entered forcing a DNA test, unless one has already been done.
DO NOT PAY ANY KIND OF SUPPORT FOR THE CHILD UNTIL A DNA TEST IS DONE, AS IT ACKNOWLEDGES YOUR FINANCIAL RESPONSIBILITY FOR THE CHILD.
Once you begin paying child support, regardless of whether you are the father or not, you can be required to continue paying.
If one has been done, go to the child support enforcement to setup a voluntary order. Wait for them to file against you will result in you paying all court costs of up to $5000
Take a certified parenting course. The court is going to order you to take it anyway, so by showing you've taken it only looks good to them.
Start keeping a daily journal of all you activities. The most common way to prevent a father from getting his rights through the courts is a false allegation. A daily journal is your number one piece of evidence in court and you can even refer to it while on the stand.
Gather evidence. Check the site below to see if it is illegal to record conversations without the other person knowing. If your state does not have a law either way, than it defaults to the federal ruling which says one person in a conversation must know they are being recorded. You're that one person. In Missouri it is specifically legal, in Kansas there is no mention either way. If you live in two different states, and one has a law against it, than it applies when the call originates from within that state,
"Can We Tape?"
Consider Bird Nest Custody. In this the child lives in one home and each parent live there for 3-4 weeks, than switch. You live with friends, family, rent a room, etc. on the off time. In this manner, your child's life is not disrupted. They are not being shifted back and forth. They keep their own room, friends, school, etc. It is hard on the parents, but than the child comes first. This is the form of custody rocker Ted Nugent had.
Bird Nest Custody
If you want to learn how to do all this go to Dads House in Yahoo Groups. There's an educational manual in the file section that can teach you what you need to know. Take the time to learn what you can and should do.
Dads House
A couple of additional
Fathers & Families
Parental Alienation Syndrome