I agree that bi-lateral healthcare arrangements exist between the UK and countries outside the EU but, in the case of EU member states, I believe they are all through EU Directives such as Directive 2011/24/EU. There are many others. For example the EHIC and Form S1 are EU wide, including EEA and Switzerland, but don't apply outside that area. Either the negotiation process would need to agree that the provisions of the relevant Directives should continue in force, possibly modified in some way, or, at the end of the process, the arrangements would cease to be valid. If an agreement on continuing these arrangements cannot be reached between the UK government and the rest of the EU as a whole then I think, under EU law, member states would not be allowed to negotiate bi-lateral deals with the UK.
The same would be true of all rights granted under EU Directives. Either they are maintained as part of the negotiating process or they cease and cannot be replaced by bi-lateral ones. As far as I am aware bi-lateral agreements in force before the EU came into existence, like the Double Taxation Agreement, can continue but member states are forbidden from negotiating new bi-lateral deals on virtually anything.
I don't think the proposed idea of allowing UK citizens rights of residence would include anything about things like healthcare even if it did come into force. The two are separate issues.