As far as I can tell, the mandate applies to employers, not individuals.
There are 2 mandates: the HHS mandate, which the Church vehemently objects to and is currently fighting in court, and the Individual mandate, which the Supreme Court upheld last summer as a tax.
The latter was not strongly objected to by the Church since the subject of individual coverage per se is mainly prudential.
It may be possible for an individual to purchase insurance without abortion coverage through the exchanges (read the policy carefully before applying - the law indicates that abortion coverage is a separate $1 charge but it is not a la carte - if the plan covers it, the insured must pay that dollar).
There is an alternative that allows individuals to avoid the non-insurance tax. Here, I quote myself from an earlier thread:
Samaritan Ministries is a non-Catholic health sharing organization. They distribute the costs of medical care among their members and negotiate fees.
Solidarity Health Share is a Catholic organization that will do the same when they get it going (so far, they have a signup form to request info when they become available). This organization was established specifically in response to the HHS mandate.
Strictly speaking, neither is an insurance company. However, the law allows participation in such programs
in lieu of purchasing insurance.
Solidarity’s main page isn’t loading for me at the moment, but their
blog still seems to be operational