Can the parameter \$\alpha\$ and \$\beta\$ be used for any BJT?
Given that \$\alpha\$ and \$\beta\$ are related by \$\alpha = \frac{\beta}{1+\beta}\$ as stated in the wiki article, obviously you can do your sums with either.

However, which is going to be easier to use? I personally always use \$\beta\$, regardless of the transistor configuration.

In common emitter \$I_c = \beta\times I_b\$, so I can say 'I need to control \$I_c\$ collector current, I need at least \$\frac{I_c}{\beta}\$ of base current to do it'.

But as \$\beta &gt;&gt; 1\$ (for most transistors), \$\alpha \approx 1\$, and \$I_c \approx I_e\$. You may object to the approximation, but given the way that \$\beta\$ varies with temperature, \$I_c\$, and between transistors of the same type, that is a far far better approximation than insisting that \$\beta\$ is constant. Any good transistor design will allow for operation with a range of \$\beta\$, at least \$2:1\$, preferably more.

Once you have made the approximation \$I_c \approx I_e\$, then common collector operation is given by 'I need to allow for a base current of \$\frac{I_c}{\beta}\$ to flow in the base circuit, without upsetting operation'. 

With a common base stage, you say much the same thing, allowing an amount of base current, however you also say that the emitter to collector gain is slightly less than \$1\$, a fraction of \$\frac{1}{\beta}\$ less than one. The error of the gain from \$1\$ will usually be a smaller error than resistor tolerances and other sources of gain error.

Given that you can write an equation for \$\alpha\$, does that mean that you need to? For most practical engineering designs, the answer is no. If you are in college, and the tutor really likes to use \$\alpha\$, then the answer is yes.