Technically yes, most amino acids will be slightly negatively charged at physiological pH. But we need to understand what that means. Let's take alanine as an example. Any individual alanine molecule can either be uncharged or have an integer net charge- it's either protonated or it's not. You will never find a "partially charged" alanine molecule. However, if you have a bunch of alanine molecules in solution physiological pH, they won't all behave in exactly the same way at exactly the same time. One molecule may have a protonated N-terminus while another has a deprotonated N-terminus. At physiological pH, the C-terminus is pretty much completely deprotonated, so we just have to pay attention to the N-terminus. We can use the Henderson-Hasselbach equation to figure out what percentage of alanine molecules will have a protonated (positively charged) N-terminus and what percentage is deprotonated. pH = pKa + log(deprotonated/protonated). If we substitute in our known numbers, we get 7.4 = 9.5 + log(deprotonated/protonated)
Therefore -2.1 = log(deprot/prot) and 10^-2.1 = deprot/prot = 0.008. That means that out of 1000 alanine molecules, about 8 of them are deprotonated and therefore negatively charged. The other 992 are zwitterions. So for all intents and purposes alanine is neutral at physiological pH.