There is no automatic entitlement to such a benefit. Any share ownership scheme should be clearly spelled out in your contract of employment as an express term. However, irrespective of what the contract states, it may be possible to establish an entitlement by virtue of custom and practice.
Also, consider whether you are being treated less favourably than other workers for a reason connected with a characteristic protected by equality legislation, such as age or sex. For example, in the circumstance where private healthcare is only offered to longer-serving members of staff who are overwhelmingly aged over 50 or male, you might well be able to lodge a claim of unlawful discrimination with an employment tribunal.
Schemes approved by HM Revenue and Customs give exemption from tax on the value of the shares at the time of acquisition, but the realising of the asset value of shares, or on termination of employment is a complex matter. You should get further advice from the local tax office.