Short Answer
One of the more famous dilemma's in philosophy is the trolley problem. One has a choice, and neither outcome is desirable. But what you present is NOT a dilemma, even if it appears to be one. In reality, it is a case of Hobson's choice, which is to say, there is not a difference in outcome regardless of what you choose. Whether someone chooses to give water or food, death is the same result.
Long Answer
Analyzing Your Argument
Your argument:
P1: X needs both food and water to survive.
P2: If X gets food, he cannot get water.
P3: If X gets water, he cannot get food.
C1: Therefore, getting both food and water is impossible for X. [from P2 and P3]
C2: Therefore, X will not survive. [from P1 and C1]
Translated into both FOL and propositional logical syntax and stripping out the modality:
P1 gets(x, food):[P] AND gets(x, water):[R] -> survives(x):[T]
P2 gets(x, food):[P] -> ( NOT gets(x, water) ):[Q]
P3 gets(x, water):[R] -> ( NOT gets(x, food) ):[S]
C1 THEREFORE ( NOT gets(x, food) ):[S] XOR ( NOT gets(x, water) ):[Q]
C2 THEREFORE NOT surives(x):[T]
What you have are two arguments. First, the argument is valid, as P2 and P3 taken together means their hypotheses might appear mutually exclusive. Hence, if P then not R and if R, then not P. This of course is exclusive disjunction P XOR R reflected in C1 which shows under either case P1 is false. Let's rewrite to simplify.
P1' gets(x, food):[P] XOR gets(x, water):[R]
P2 gets(x, food):[P] -> ( NOT gets(x, water) ):[Q]
P3 gets(x, water):[R] -> ( NOT gets(x, food) ):[S]
C1 THEREFORE ( NOT gets(x, food) ):[S] XOR ( NOT gets(x, water) ):[Q]
Now, this sure looks like the constructive dilemma, but is it? Let's rewrite it as a constructive dilemma.
P1' gets(x, food):[P] OR gets(x, water):[R]
P2' gets(x, food):[P] -> ( NOT gets(x, water) ):[Q]
P3' gets(x, water):[R] -> ( NOT gets(x, food) ):[S]
C1' THEREFORE ( NOT gets(x, food) ):[S] OR ( NOT gets(x, water) ):[Q]
Well, from a strict syntactical perspective, we can say they are different. Obviously exclusive and inclusive disjunction have different truth tables and are not logically equivalent. But, we can still ask, is your argument still some form of dilemma, and the answer is no.
Why It Is Not a Dilemma
Your argument seems to have two possibilities, that a person might get food or might get water, so why isn't it a dilemma? Because a dilemma has two necessary conditions. There must be a choice, but there must be different, undesirable outcomes. In your argument, there is NO difference in outcome, so while you can choose one case or the other, the outcome is the same. This is known as Hobson's choice.
From WP:
A Hobson's choice is a free choice in which only one thing is actually offered. The term is often used to describe an illusion that multiple choices are available. The most well known Hobson's choice is "I'll give you a choice: take it or leave it", wherein "leaving it" is strongly undesirable.
Remember, from "dilemma" (WP):
A dilemma (Greek: δίλημμα "double proposition") is a problem offering two possibilities, neither of which is unambiguously acceptable or preferable. The possibilities are termed the horns of the dilemma, a clichéd usage, but distinguishing the dilemma from other kinds of predicament as a matter of usage.1
Are there two possibilities in the outcome of your argument? If one is referring to undesirable outcomes, then 'no'.
The Psychology of Dilemmas
So, if the argument was that giving a person food allowed them to live albeit losing their right leg and arm and giving them water allowed them to live albeit losing their left leg and arm, you might have a bit of a dilemma, because the choice of giving food or water creates a psychological stress knowing that both choices do harm. But in your argument, both choices lead to the same outcome, so an indifference can be had since neither choice really has an impact on the outcome.
While it might be common to think of both the constructive and destructive dilemmas as logical forms, as Jo Wehler pointed out in his response, the root of the dilemma is human emotion, and the term was used long before symbolic, formal systems systematized the study of reason. From his answer:
In classical Greek literatur [sic] a dilemma is a situation where a person has to take a decision between two alternatives. In both cases the person becomes guilty.
The prototype is the dilemma of Antigone: She has to make a decision between the law of her city and a religious law of her society. The former forbids, the latter demands the same action.
Lastly, note that this is the danger of symbolic forms, that it becomes possible to lose sight of what the form represents. It always behooves a critical thinker to remember that syntax is an effective, but sometimes shallow substitute for semantics.
