Can't recommend Basel Unispital highly enough. Our baby was born there August 2009. Great rooms, fantastic staff (really caring and supportive).
Only down side... there is no A/C in the individual rooms (just in birthing rooms) so if you are having a baby in July/August it is hot in the rooms. Good for baby... not so good for mother.
About having private insurance.... It really depends on availability. Last year was a record year for births in Basel. At certain times people with private/full insurance could not get a private room because all rooms were full. Mind you if you can't get a private room you will only be sharing with one other person..max.
Good Luck with the baby plans!