Do you mean like a baby bouncing chair? I have never heard anything such as this. My mom used baby rocking chairs for all 9 of her children including me and my two sisters who have PKU. We have never had any issues because of it.
Also what this dietitian told you about your baby not getting enough protein is technically not true. Although we do not eat protein from natural sources such as meat, dairy, grains, and so on we do get protein from our formula. If you look on the label of your babies formula it actually contains quite a bit of protein BUT it does NOT contain phenylalanine (which is the amino acid people with PKU cannot have). Our formulas contain all (or most) of the amino acids that make up protein except phenylalanine.
Your son will be fine in his baby rocking chair! I'm not sure why this dietitian told you this.