I think this question has been addressed many times on Cross Validated: proposing a negative value when the target distribution is of no consequence since the target density is equal to zero at this negative value. A Metropolis-Hasting algorithm that is described in the question as a random-walk version of MH thus works in such a context: for each proposal with a negative variance, the chain repeats its current value.

See, e.g., 

 - [MCMC Metropolis-Hastings' jumping distribution for non-negative
   parameters][1]
 - [Narrow distribution and Hasting-Metropolis][2]
 - [Elementary MCMC pseudocode][3]


  [1]: https://stats.stackexchange.com/q/260230/7224
  [2]: https://stats.stackexchange.com/q/180372/7224
  [3]: https://stats.stackexchange.com/q/47459/7224