Really it's going to depend a lot upon the amount of bulbs you have in the unit. Some CFL softbox units only have a couple of bulbs. Some have considerably more. Among all the studio gear I own is a Lencarta Quadlite, which as the name suggests, has 4 bulbs inside. That does put out a very reasonable amount of light. Of course the distance the light is to your subject will affect exposure greatly, so getting it as close as you can without it being in the frame would be advisable. You also have the option to raise your ISO, with most modern DSLR capable of getting clean images at ISO 800 to 1600. Older DSLR like the D90 and below, I wouldn't venture above ISO 800.
With all that said, if it's for newborn photography, generally the kids are asleep or at least not exactly jumping about the place, so you aren't going to require a particularly fast shutter speed. If you're hand holding the camera aim to get a shutter speed equivalent to your focal length + any crop factor. If you're tripod mounted then even very slow shutters can be achieved with the use of a release cord or wireless remote. Alternatively using a timer. Which ever you use, enable the mirror lock up or exposure delay to avoid mirror slap blur. If you're shooting with a 35mm or 50mm lens, then achieving a blur free shutter speed shouldn't be a problem even at native ISO (usually 100-200).
I use the YN560 II strobes myself in studio and they are great value for money. However, for newborns I wouldn't use flash. Mainly because of disturbing the child if it sleeps. Continuous light is something they will much more readily accept, than a strobe going off. One major bonus of a CFL light or a monolight (PCB Einstein, Alien Bee, Elinchrom, Bowens, etc), is the ability to see in real time, exactly what light and shadows are doing. If you are using strobes like the YN560, you will have to chimp your exposure in the absence of a light meter, as they aren't TTL. Yongnuo do make TTL strobes for a few bucks more. The downside to speedlite type flashes is that you won't see what the light is doing, until you have taken a picture. This will mean the flash firing a few times to fine tune your light, which can be a disturbance to a new baby. If you do go with strobes, be sure to turn off the recycle beep.