Being a kitten, he'll have a lot of excess energy to wear out. Was he making the odd noises before he began being sick? If so, then it's probably just play and excitement, but if the noises and squeaks have only begun since he showed signs of sickness, it may be a sign of something stuck and causing pain or discomfort, either in his throat, stomach or bowel.
Try to watch when he begins to dash about, and see if it seems like normal play, or whether he's trying to "escape" something. Often, if it's the latter, you might catch him, biting at a limb, or his tummy, or shaking their head or nudging at their bottom. It's not unknown for young cats and dogs to ingest things they shouldn't - needles and threads for example, bits of wool or string or plastic, those kinds of things. If the squeaks appear to indicate agitation, then I would definitely get him checked over.
