Bill, sometimes the spikes don’t round out on some fruit. If the fruit have been red for a couple weeks, it should be ripe. Usually the smoothing out of the spikes is an accurate way to tell if it’s ripe but some of the fruit with small or aborted seeds sometimes don’t follow this rule.
Your fruit looks very round and fat for a Brewster. Brewsters your size should produce a lot more fruit. Without knowing your fertilizer regimen, I would hazard to guess that your Lychee is receiving a bit too much of Nitrogen.
Right now, after you harvest your fruit, is a great time to prune your tree so that it can push a vegetative flush, have it harden and be ready for the winter.
Simon