The company can pay for health insurance and get tax relief but it is taxed on you as a benefit in kind. That means if the company paid £1,000 for your health insurance then you’d have to pay £400 personal tax on the benefit (assuming you’re a higher rate tax payer in the year)
The company also needs to pay Class 1A National Insurance of 13.8% on the premiums.
And finally… you’d need to submit P11D forms to report it to HMRC or pay an accountant to do the forms for you.
All that said it’s not necessarily a bad thing. Sometimes the rates that insurers will do for companies are lower than they’d do for you personally.
Good insurances to put through the company are life insurance and executive income protection. Both of these are tax deductible in the company but not taxed on you.
Executive income protection is there to pay out if you can’t work for some reason (serious stuff rather than a head cold). It’s always worth having a think about how long you could financially survive for if you were unable to work.