Value Killers
Altruism
Defeat this one by claiming the Egoist standpoint. This brings out that, other than that it's impossible, altruism is not a moral model but a model for the Red Cross credo. Try to show one of two things: 1) your opponent's position attains a self-serving goal of greater magnitude, or, if they claim the side-effect of benefiting oneself is immaterial, 2) show how, separating the self-benefit it provides, your value is Altruistic as well.
Autonomy
To be without outside influence leaves a person or nation to infringe on any number of other values, including freedom, liberty, natural rights, and individualism.
Communitarianism
Who determines what is best for the community? What makes one set of ethics better than another? How is it possible to achieve moral absolutes? If they know their Sandel and Etzioni and have an answer for this one, swing away. Hit them with the flaws in the inter-community moral disputes. Read Sandel and Etzioni to know what's missing, then read some Indiviualist critiques to see how to say what's missing.
Dignity
If dignity is inherent, it can never be removed, and is, therefore, also inherent in all other values. If Dignity can be removed, then it isn't inherent, as it must then be recognized by outside forces to exist, and it becomes a relative value. Saving the Dignity of 100 people is more moral than saving the Dignity of 10 people.
Equality
Same thing. Equality isn't always moral. Should we all be equally dead, deaf, dumb, blind, ugly, etc?
(Ethical) Egoism
The idea that since self-interest influences all actions, it must be a moral value is flawed. Actions motivated by self-interest are not always moral (i.e. murder, robbery, rape), and if the self-interest is governed by ethics, it is still relative to personal ethics, not absolute morality. Therefore, not everyone will act morally (i.e. pathological killers, liars, etc.)
Freedom
"without arbitrary restrictions." What is arbitrary? If everything is, than I can kill you. If some things aren't, than how is it freedom if you have to sacrifice individual rights? If they answer that one, go for the jugular, claim their value doesn't clash, if they're neg, or is non-unique (applies to both sides of the res), if they're aff. Also point out that you are not forcing an arbitrary restriction and are therefore also achieving the value of Freedom.
Individualism
Me First! Me First! That's what it will sound like to the judge if you claim that this value your opponent is upholding is sacrificing all others for the sake of the individual. Do we really have to first look out for #1? Running a Hobbes-ian or Rousseau-ian critique of the nature of man and the nature of the state might be good. Read both SC's to get that info.
Justice
Easy. What is just is not always moral. Think of it using the square-rectangle metaphor. A square may always be a rectangle, but a rectangle isn't always a square.
Liberty
Harder to defeat. The liberty concept can be defeated with the Communitarian or Altruistic idea that liberty sacrifices personal pleasure for overall progress/gain/improvement. You'll just have to outweigh it.
Natural Rights
Very hard to defeat. Only way to beat this is to either incorporate them into your own value or claim that a greater moral good is achieved by sacrificing them on occasion. Use Mill's limitations to rights to your advantage. Know Mill and Locke. Try to get them to abide by the Lockean Theory of NR. Then attack them with the paradox of Locke's property theory. Read some critiques of Locke to get this one.
Quality/Sanctity of Life
Life doesn't always win. This one is about ends and means. Q/S of L is the beginning, but almost any other value will come after the attainment of Q/S of L, and, therefore, already incorporate it in the other value.
Societal Progress
Progress at any cost? If so, stampeding natural rights is included, and that's obviously immoral. If not, certain values must be superior to it and you will have them at your disposal.
Stability/Security
Sacrificing freedom, liberty, and natural rights is necessary for stability/security. The logical extreme of this sacrifice is 1984. Even if you don't buy that, and you claim citizens of a nation tacitly consent to the sacrifice through the Social Contract, achieving Stability/Security requires infringing on the same rights of individuals not bound by the SC. This sacrifice is immoral.
Utilitarianism
Greatest good for the greatest number? Let's just stampede individual rights. Can you say Hitler? You can, if your opponent or judge looks ignorant. If not, I suggest going a little deeper into the theory of Spinoza and Mill. They outline what Utilitarianism is. Look for holes in the Higher and Lower Order Pleasure Theorem. If you're still lost, read Mortimer J. Adler's Ten Philosophical Mistakes. He outlines the problems with Utilitarian views of relative morals.