According to the manual class fields are private by default. However when converting to target java, jvm or cs, the resulting fields are public. (BTW even if I declare them private, the resulting fields are public.)
How can I make them really private?
You can decorate the fields with metadata. See haxe --help-metas:
@:private : Marks a class field as being private. (cs only)
@:protected : Marks a class field as being protected. (for flash,java,cs)
As for the comments on which platforms do support it, it’s best to actually check 
Checking haxe --help-metas I have found that there exists also the metadata @:internal for the targets java and cs. Unfortunatley @:private is only for cs.