Research:

Few studies have been published about the effectiveness of resistance bands, though several have shown that explosive lifting (which bands are) is more beneficial than traditional, slower resistance training.  Out of the studies that have been done on bands specifically, benefits have been observed.

Studies Done on Bands Similar to the BNS Bungee Bands:

One study looked at electromyography (I-EMG) activity while using bands and chains as 10% of the overall resistance (1).  They concluded that there was no difference between using or not using chains and bands in terms of muscle activity, and concluded that bands and chains were troublesome to set-up.

All other studies, however, have shown that bands are effective, and the authors suggest that they are quick and easy to set-up. 

Another study looked at the effects that elastic bands had on force, power, and velocity in the back squat (2).  They discovered that bands, when applied to produce more tension at the top of the lift, improve velocity and power in the bottom portion of the lift using a 6 repetition-maximum (6RM) resistance.  A person's 6RM is usually 80-85% of a person's 1RM.  The authors concluded that bands may be helpful in training for vertical jump or other ballistic movements.

A more recent study done at Cornell University (the author asked me not to give the complete reference until it is published) compared the effects of using BNS Bungee Bands versus regular strength training on squat, bench press, and vertical jump power, in addition to increases in lean body mass.  The program lasted for eight weeks, and both groups used the same lifting protocol.  The group that used the BNS Bungee Bands improved significantly more than the control (traditional weight training) group in all of the variables. 

The most recent study (3) looked at the effect that using the BNS Bungee Bands at different percentages of 1RM had on maximum force, power, and rate of force development, compared to not using bands.  The authors discovered that using bands at 60% of a person's 1RM back squat (see "Different Kinds of Bands" page for how to calculate resistance) did not impact force, power, or rate of force development when compared to not using bands at the same resistance.

However, when using bands at a percentage of 85% of a person's 1RM, maximum force and power were shown to be significantly higher.  This was true for for both testing protocols, which used bands as 20% of the overall resistance, and 35% of the overall resistance.  The authors of this study suggest that bands may be a good way to increase both strength and power at the same time using at or near 85% of a person's 1RM, compared to needing to train these two characteristics during different workouts, or even different training cycles.

Studies Done on Bands Similar to the Squat Sling bands:

The Newton study also looked at bands providing "spring" at the bottom of the squat.  They observed no differences in velocity, power, or force in using the bands in this manner compared to the traditional squat. 

A preliminary study which actually used the Squat Sling bands showed that power is unchanged compared to the traditional squat, but that peak force was higher.  The resistance used was approximately 80% 1RM, with approximately 24% of the resistance coming from bands.  The results of this show that perhaps this type of bands may be useful for training strength.  

As of now it is unclear as to whether or not the Squat Sling bands are an effective means of increasing power, force, or velocity compared to a traditional barbell squat.

References:

1) Ebben, W., Jensen, R. 2002: Electromyographic and Kinetic Analysis of Traditional, Chain, and Elastic Band Squats. The Journal of Strength and Conditioning Research: Vol. 16, No. 4, pp. 547–550.

2) Newton. 2002: Heavy Elastic Bands Alter Force, Velocity, and Power Output During the Back Squat Lift.  NSCA Conference.

3) Wallace, B., Winchester, J., and McGuigan, M. 2004: Effects of Elastic Bands on Force and Power Characteristics During the Back Squat Exercise. Presented at the 2004 NSCA National Conference - Minneapolis, MN.