Unfortunately, if your boyfriend has signed the birth certificate and an aknowledgement that he is the father and the child is his, he does have legal obligation to pay child support for this child and if he is not paying child support that has been ordered, he would be subject to the child support enforcement, which can subject his driver license and US passport to being revoked and also his wages and assets to being garnished to satisfy the back owed child support.
Since you boyfriend never contested the paternity withing the first year after aknowledging it and signing the birth certificate, it is not likely that he would be able to dissestablish paternity, even if he turns out not to be a biological father, due to the lapse of time and the only option for your boyfriend to get custody and visitation of his minor child is to file Petition to Establish Custody and Visitaiton in a Family court
in Oregon and the court will review your boyfriend's petiton and upon the full review of the case the judge will decide the amount of custody and visitation that your boyfriend would be awarded.
Unfortunately, this is a complex legal field and it would make sense for your boyfriend to consulty and retain local child custody attorney to represent him, if he does decide to file petition for custody / visitation.
I am very sorry to provide you with this tough news, but unfortunately visitation has nothing to do with child support, as they are considered two separate legal issues and have no connection to each other.
I wish you and your boyfriend the best of luck!