1. What you need to do is to apply for a Permanent Residence card once your six month EEA Spousal visa is going to expire. Here is a link to the form https://www.gov.uk/government/publications/apply-for-a-document-certifying-permanent-residence-or-permanent-residence-card-form-eea-pr . You can then use this Permanent Residence card for the next five years to certify that you can live and work in the UK. In essence, it merely evidences your next five years on the EEA Spousal visa in the UK. the fact your husband is sick will not affect the application for the Permanent Residence Card.
2. At the end of five years in the UK, you can apply for UK citizenship. However, you should have had private health insurance for those five years in order to be able to gain Uk citizenship. So, if you don't have it now, you should get private health insurance.
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