The prior is not (yet) at issue: A simple linear regression doesn't appear an appropriate choice for your data. More plainly, that the model could produce negative predictions of positive data shows an improper choice of likelihood.
What it seems you're looking for is a generalized linear model. In brief, a simple linear regression assumes that $y|x \sim N(X\beta, \sigma)$; in words, that $y$ is normally distributed around some linear function of $X$. A generalized linear model treats the data as produced by some other distribution, and transforms this linear function to produce a parameter for that distribution. Logistic regression is a familiar example: The data are Bernoulli generated, with $p = S(X\beta)$, where $S$ is the sigmoid function.
For more on this Andrew Ng's introduction is nicely approachable, and a fuller treatment is available here. For continuous, positive data, typical choices are the gamma and exponential distributions. From Ng's tutorial, emphasis mine:
The multinomial (which we’ll see later), the Poisson (for modeling count-data; also see the problem set); the gamma and the exponential (for modeling continuous, non-negative random variables, such as time intervals); the beta and the Dirichlet (for distributions over probabilities); and many more.
Modeling your data using an appropriate generalized linear model with appropriate priors, rather than a simple linear regression, would solve your problem of negative predictions: All your $y$ values would be positive, because they would all be sampled from a distribution with strictly positive support.
