Should Company B enter the market in the following cases
Knowing that demand is linear, we can deduce from the two points on the demand curve, $(q,p)\in\{(1,2),(2,3)\}$, that the demand function is $p=7-2q$, which is consistent with the Company A's profit maximizing quantity being $2$ units (assuming no fixed cost). 

If B enters the market and competes in quantity, the Cournot outcome is $q_A=q_B=\frac23$ and $p=\frac{13}3$. Taking into account B's fixed cost of $1$ for entering the market, its profit is 
\begin{equation}
\left(\frac{13}3-3\right)\frac23-1=-\frac19&lt;0.
\end{equation}
Hence it's not profitable for B to enter. 

If B enters the market and competes in price, the Bertrand outcome is $p_A=p_B=3$. In this case, revenue is equal to the cost of production, so B makes zero profit (absent the entry cost). However, given the positive cost of entry, B would be making a loss if it enters the market.