Note that shares are artificial concept, present in mining pools. In the Bitcoin network, you have valid blocks, or invalid blocks. Any block that does not meet the target is considered invalid, even if all included transactions are valid. Miners are rewarded only by mining pools for those invalid blocks, because they meet some lower difficulty, set by that mining pool.

Note that there are also non-mining nodes, and using Proof of Work to filter connections will eliminate them. The question is: do you want to also eliminate for example block explorers, if their operators will not start mining?