I don't think there are any concerns with Java versions based on the particular flavor of Windows 7 that you are running. Both the JDK and JRE should run fine, and both versions 1.6 and 1.7 should work.
System Manufacturer/Model Number Self-Built in July 2009 OS Windows 7 Ultimate x64 CPU Intel Q9550 2.83Ghz OC'd to 3.40Ghz Motherboard Gigabyte GA-EP45-UD3R rev. 1.1, F12 BIOS Memory 8GB G.Skill PI DDR2-800, 4-4-4-12 timings Graphics Card EVGA 1280MB Nvidia GeForce GTX570 Sound Card Realtek ALC899A 8 channel onboard audio Monitor(s) Displays 23" Acer x233H Screen Resolution 1920x1080
Keyboard ABS M1 Mechanical Mouse Logitech G9 Laser Mouse PSU Corsair 620HX modular Case Antec P182 Cooling stock Hard Drives Intel X25-M 80GB Gen 2 SSD
Western Digital 1TB Caviar Black, 32MB cache. WD1001FALS Internet Speed 15/2 cable modem Other Info Windows and Linux enthusiast. Logitech G35 Headset.
System Manufacturer/Model Number HP pavilion DV6-Laptop OS win7 home premium-64bit-SP1-IE9RTM CPU T6600 2.2Ghz Motherboard HP Model 3628 Memory 4 Gb Graphics Card ATI Mobility Radeon HD 4530 Sound Card IDT High Definition Screen Resolution 1366x768 @ 60Hz