If strict convexity of indifference curves isn't assumed, does MRS have to be negative? Strict convexity is defined as 


  Let $X$ be a convex set in a real vector space and let $f: X\rightarrow \Bbb{R}$ be a function. $f$ is called strictly convex if $\forall x_1 \neq x_2 \in X,$ and $\forall t \in (0, 1)$:$$f(tx_1+(1-t)x_2) &lt; t f(x_1)+(1-t)f(x_2)$$


If this doesn't hold, does marginal rate of substitution still have to be negative?
The marginal rate of substitution is the slope of the indifference curve multiplied by -1. Since the indifference curve is negatively sloped (as soon as the person prefers to consume the goods), the MRS is always positive. This has nothing to do with the convexity of the preferences.

If the indifference curve is convex without being strictly convex, then it is linear, i.e. it has constant slope. In that case, the marginal rate of substitution is constant (but still positive!).