You can have authoritarian left and authoritarian right. That's not really related to healthcare. Authoritarians tend towards abuses of power - I kind of don't find it terribly relevant if millions were killed or imprisoned because of the differences in their ideals, which is what the horseshoe theory speaks to. What it speaks to is mainly that the methods - e.g. unconstrained power of an executive or government, become more similar than not as extremism increases. It's not really saying that the ideology necessarily comes to match, only that the ideology matters less than whatever horrible things are coming your way, beliefs be damned.