The current through a BJT is controlled by the base-emitter voltage VBE, it depends exponentially on that voltage. From this point of view the BJT is a voltage-controlled device.
The way a BJT works makes it necessary to also have a small current through the base. This base current is only a small fraction of the collector current. The ratio between these currents can be used to define a current gain and it is possible to regard the BJT as a current-controlled device.
So we have two ways to deal with a BJT and depending on the situation one is better than the other.
For a CB circuit we see that input and output current are almost equal and it could be used as a current buffer. Apart from that current gain doesn't help much to gain further insight.
Using the voltage-controlled operation approach, we see that the emitter gives direct access to VBE of the transistor in almost the same we as for the CE configuration, only the sign is different. So a change of the emitter voltage causes a change of VBE. This also changes the current through the transistor which is then converted into a voltage by Rc. So it is possible to achieve voltage amplification.
However, usually voltage amplifiers with a larger input impedance are wanted and therefore the CE stage is preferred over the CB configuration. This leaves the use as a current buffer like in a cascode circuit.
Given your instructor's remark, I suspect that you were actually told about "debouncing", which a very useful function. However, the circuit shown is a bad, bad example of how to do it.
Let's think about a switch making contact. Inside, you have two pieces of metal making contact, and on a timescale of milliseconds the two pieces can actually bounce, perhaps repeatedly, before settling down. If this happens the switch output will apparently show multiple activations where only one was intended. To combat this, a circuit like this can be used:
simulate this circuit – Schematic created using CircuitLab
You've noticed that it has the same general outline as your circuit, but a few more resistors.
Let's say that the switch is open, and gets briefly closed, then open, then closed for good (a single bounce). Current through R1 turns on Q1 and pulls the collector low. This also discharges C1. As a result Q2 is turned off and Vout goes high. When the switch bounces, Q1 is briefly turned off and its collector rises. However, the resistor charges up much more slowly than it discharges (for appropriate resistor values of R2 and R3), so Q2 never turns back on. As a result, the output Vout has been "debounced", and R2/R3/C1 can be selected for any desired bounce time to be ignored.
The original circuit is not very good, since the capacitor voltage swing is quite small, due to the clamping effect of the Q2 base-emitter junction.
Best Answer
"My question is that is it possible for a BJT VCVS to have a gain > 1. Thanks."
I suppose, with your question you refer not only to the shown circuit (common collector configuration). Thus, my interpretation of your question is as follows: "Is it possible to realize a VCVS using one single transistor?".
My answer: A device which we call VCVS must have a very large (ideal infinite) input resistance and a very small (ideal: zero) output resistance. Both requirements cannot be fulfilled with a transistor. However, the shown circuit can be a rather good approach because we can realize an input resistance in the order of some hundreds of kohms and an output resistance of 50 ohms or so. However, as was mentioned already, the gain of this circuit cannot be larger than unity.
Of course, there are other circuit alternatives (common emitter) with gain values that are much larger - however, the required characteristics for a VCVS (input, output resistances) cannot be fulfilled as we could for the common-collector circuit.
As a consequence, if you need a "good" VCVS with gain>1 you need 3 transistor stages in series: : A gain stage (common emitter) and two other stages (common collector) at the input as well as at the output of the gain stage.