Research over the years has shown that insects including mosquitoes cannot transmit HIV. HIV cannot live inside of an insect and therefore cannot be transferred from insect to human.
No, you cannot get AIDS or even hepatitis from insect bites. The only risks are Lyme Disease (from chiggers and ticks) or West Nile Virus (if from mosquitoes). AIDS is only a concern if you share needles, have sex, or other bodily fluid exchange with someone who is infected with the virus.