Integrity
If Jim killed one of the Indians, wouldn't he be a murderer? Why should Jim be morally obligated to do something evil? Also, if murder is acceptable in certain cases, is everything acceptable? (see the "Possible Exceptions for Everything" section)
Jim feels stressed and worried about killing, since he feels it is wrong, doesn't that mean he would be violating his personal integrity if he killed one of the Indians? No moral system should ask one to violate their integrity (see the "Distastefulness" section).
Jim's uneasiness to do something difficult to help the villagers and the Indians is not what forms his integrity. Refusing to act whenever challenges arise is the opposite of integrity as it means one would not hold to any moral principles except when life is easy. Jim's integrity would not be violated if he held to the foundational principles he believes in. If those were utilitarian principles, he would be holding to moral principles which ask him to do what he can to best help others, even if it causes some pain (including Jim's own uneasiness) for the benefit of great, long-lasting happiness.
If Jim's uneasy feelings were the basis of the ethics he followed, then what value are those ethics which would be based on whatever he felt like? If ethics did not require one to change their personal beliefs to pursue higher moral ends, then what value does ethics have? If one's personal opinions about ethics are flawed, it is their responsibility to change their opinions (see the "Distastefulness" section).
Distastefulness
I don't Like your Ethics
General Distastefulness
Most criticisms are caused by how utilitarianism ethics do not match one's current thinking. This is not a problem with utilitarianism, this is a problem with ethics – ethics asks people to change their thinking and actions for the sake of doing what is right. It would be concerning for an ethical system to base itself on one's individual impressions and judgements.
Other criticisms seem to caused by complete misunderstandings – misinterpretations of utilitarianism principles which no one should ever approve of. Every ethical system experiences this and the solution is to calmly and rationally listen to what that ethical system actually teaches. Remember, utilitarianism's goal is to help everyone be the happiest they can be. Is an honest, wholesome pursuit of happiness a bad thing?
Common Criticisms
Criticism: Emotion is required to make good judgements. Impartiality seems to make people emotionless and therefore causes bad judgements. A common example, helping someone in great need over helping family in some need.
My response: Is it not concerning to prefer not helping people in the greatest need? Utilitarianism does rely on emotion, it relies on pure love for all others. This pure love has been integral to utilitarianism since its foundation: "to do as you would be done by, and to love your neighbour as yourself, constitute the ideal perfection of utilitarian morality" (John Stuart Mill, Utilitarianism, Chapter 2).
Criticism: Consequentialism seems to encourage people to choose morally unthinkable actions.
My response: Most example criticisms are complaining about making the best of a bad situation and not about how utilitarianism does so. A common example is choosing not to try to save someone in danger when the danger is still present since it means abandoning someone in their time of need. Such an example often ignores how saving them is more likely to end up injuring one's self, causing the would-be rescuer and the other person to both require rescuing – worsening the problem. Unfortunately, a bad situation generally means bad things will happen. Consequentialism just tries to minimize the damage. Is that not what anyone should do?
Criticism: Hedonism seems to not care about rights. It should support deontological ideals.
My response: Why are rights, freedoms, etc. good? Because they provide happiness? Since they give pleasure and lessen pain in the world? Those reasons are why hedonism supports rights, freedoms, etc. (see the "Do Rights Matter?" subsection of "Possible Exceptions for Everything").
Blind Obedience
Your Ethics are simply Rules to Follow Robotically
Ethics =/= Mathematics
Trying to calculate or estimate the moral correctness of an action is trying to turn ethics into mathematics.
Ethics requires judgement, which utilitarianism seeks to teach. Judgement (not mathematics) will always be required for any moral scenario. Jeremy Bentham did create a form of calculus (hedonistic calculus) meant to help estimate what might be better decisions. It is an optional tool one may use – nothing more. Concerning it, he taught, "it may, however, be always kept in view" (Jeremy Bentham, Of the Principle of Utility) rather than strictly applied.
Does Consequentialism Care about Character?
Consequentialism is about good actions – not good people. Your ethics seem to be creating better robots, not better people.
Good people are those who tend to do good actions with good motives. Whether a good person or an evil person does something evil, the action is still evil. The moral of the action is what consequentialism is concerned with.
Good character is essential – it is part of how one becomes happy. Utilitarianism fosters pure love for all others, teaching that a true desire to help others as best as one can is the possible attribute one may obtain (be happy by making others happy). This is not something which may be achieved through strict obedience or calculations. Remember that utilitarianism teaches, "It is impossible to live a pleasant life without living wisely and honorably and justly, and it is impossible to live wisely and honorably and justly without living pleasantly. Whenever any one of these is lacking, when, for instance, the man is not able to live wisely, though he lives honorably and justly, it is impossible for him to live a pleasant life" (Epicurus, Principal Doctrines, 5).