First we need to define the elasticity of substitution $\sigma$. This can be a difficult and confusing concept. (If you want to have your mind blown, look at Table 2 in this survey by Stern, which classifies no fewer than 10 notions of the elasticity of substitution!).
That said, the formula that your professor mentions is true only when there are two goods, so I'm going to assume we're restricting ourselves to the two-good case - and there, defining the elasticity of substitution is (for the most part) unambiguous. I'm first going to define it as (minus) the inverse elasticity of the ratio of marginal utilities of $x$ and $y$ to the ratio of their quantities, holding the overall level of utility constant at some $u$:
$$\sigma\equiv - \left.\left(\frac{\partial \log(U_x/U_y)}{\log(x/y)}\right)^{-1}\,\right|_{U=u}$$
This measures the curvature of the indifference curve between $x$ and $y$: the higher $\sigma$ (i.e. the more substitutable $x$ and $y$ are), the closer the indifference curve is to a straight line.
Now, Hicksian demand minimizes expenditure subject to being on an indifference curve. This involves setting the ratio of marginal utilities equal to the ratio of prices: $$U_x(x^H,y^H)/U_y(x^H,y^H)=p_x/p_y$$
Hence we can reinterpret $\sigma$ as (minus) the elasticity of relative Hicksian demands with respect to relative prices:
$$\sigma=-\frac{\partial \log(x^H(p,u)/y^H(p,u))}{\log(p_x/p_y)}\tag{1}$$
Since the denominator has only the relative price $p_x/p_y$, it doesn't depend on how we change this relative price (say, by raising $p_x$ versus lowering $p_y$). Let's assuming that we're just raising $p_x$, and then use your $\varepsilon$ notation for elasticities for simplicity:
$$\sigma=-\frac{\partial \log(x^H(p,u)/y^H(p,u))}{\log(p_x)}=\varepsilon^H_{y,p_x}-\varepsilon^H_{x,p_x}\tag{2}$$
At this point, we bring in a simple identity for Hicksian demand, namely that the sum of elasticities of goods with respect to some price, weighted by their shares, is zero (proof included below):
$$s_x\varepsilon^H_{x,p_x}+s_y\varepsilon^H_{y,p_x}=0\tag{3}$$
Since $s_x=1-s_y$, we can rewrite (3) as
$$(1-s_y)\varepsilon^H_{x,p_x}+s_y\varepsilon^H_{y,p_x}=s_y\cdot(\varepsilon^H_{y,p_x}-\varepsilon^H_{x,p_x})+\varepsilon^H_{x,p_x}=0\tag{4}$$
But the term in parentheses in (4) is just the elasticity of substitution $\sigma$, as we showed in (2). Rearranging, we've proven your professor's statement
$$\varepsilon^H_{x,p_x}=-s_y\sigma$$
as desired.
Proof of (3). We know that holding income constant, Marshallian expenditure must remain constant after any change in prices. The elasticity of spending on $x$ with respect to $p_x$ is $1+\varepsilon^M_{x,p_x}$ (combining price and quantity changes), and the elasticity of spending on $y$ with respect to $p_x$ is $\varepsilon^M_{y,p_x}$. Weighted by initial shares, these must sum to 0:
$$s_x(1+\varepsilon^M_{x,p_x})+s_y\varepsilon^M_{y,p_x}=0$$
Substituting the elasticity Slutsky equation, we get
$$s_x + s_x(\varepsilon^H_{x,p_x}-\eta_xs_x)+s_y(\varepsilon^H_{x,p_x}-\eta_ys_x)=0$$
which, once we use the fact that $\eta_xs_x+\eta_ys_y=1$ (as income expands, total spending expands proportionately), has a $-s_x$ canceling out the $s_x$ at the front, and reduces to just (3):
$$s_x\varepsilon^H_{x,p_x}+s_y\varepsilon^H_{y,p_x}=0\tag{3}$$
as desired. This is a very important identity governing Hicksian demand: it says that as we change prices, to first order the cost of Hicksian demand at the old prices stays constant (this is related to it being "compensated" demand: locally, it would cost the same at the old prices).
