I'm going to assume a quarter pound (113 g) burger with no cheese or other added feature. This being the case the bread can be neglected as the water-cost of beef is going to dominate. I'm also assuming that the unit of volume being used is the American gallon and not the imperial gallon.
The answer is definitely going to depend upon how you raise the livestock as grass-feeding requires much less water than grain-feeding. That being said there is research out there on this. Estimates vary quite widely as you might expect given the in-built problem of how the cattle are farmed. The results per burger (to the nearest whole number):
- 111 gallons (418 litre)
- 480 gallons (1816 litres)
- 405 gallons (1534 litres)
- 621 gallons (2352 litres)
- 616 gallons (2331 litre).
 is a global average estimate, ,  and  are based on American data and  is based on Japanese data.
So, although I'm not sure where the number 660 gallons (~2500 litre) is coming from, it appears to be in a believable range. Taking an average of these numbers gives you 447 gallons (1690 litres). The vegan sites' estimate is close to the upper values of this group of studies. Maybe a bit high.
According to Waterwise (a UK-based NGO dedicated to water efficiency and conservation) a normal shower uses 2.05 American gallons/minute (7.75 litres/minute) and a power shower uses 4.49 American gallons/minute (17 litres/minute). Using their 660 gallons estimate, in an average month (30.42 days) of once daily showering, if you use a power shower and you have a shower for less than 4.83 minutes, you use less water than when you eat a hamburger. If you use a normal shower you can enjoy rather longer daily ablutions; anything less than 10.58 minutes will mean you have used less water than eating a hamburger. For 2 months half those times. I suppose it is plausible to have a daily 5 minutes with a normal shower. That would mean using less water use in 2 months of showering than in eating a hamburger.
 Beckett, J. L., and J. W. Oltjen. (1993) 'Estimation of the water requirement for beef production in the United States.'
The model estimates 3,682 L of developed
water per kilogram of boneless meat for beef cattle
production in the United States.
 Chapagain, A.K. and A.Y. Hoekstra (2003) ‘Virtual water trade: A quantification of virtual water flows between nations in relation to international trade of livestock and livestock products’
Table 4.4 says the world average of Virtual water content is 15206 m3
Bovine meat and meat offal not else specified,
excluding livers, prepared or preserved
 Zimmer, D. and Renault, D. (2003) ‘Virtual water in food production and global trade: Review of methodological issues and preliminary results’ (section 5)
 Oki, T.; Sato, M.; Kawamura, A.; Miyake, M.; Kanae, S., and Musiake, K. (2003) 'Virtual water trade to Japan and in the world' (section 15):
 Kreith, M. (1991) 'Water inputs in California food production'
 Water Wise