Here is a flow chart of the forms of energy, with links .
Conservation of energy is one of the fundamental laws governing physical systems and is the only reason why one can talk of "negative energy"
here is a breakdown of the forms that **Conservation of energy ** takes
In almost all frames negative energy exists, in the sense of conservation of energy, for example between potential forms and kinetic forms. Look at the energy levels of the hydrogen atom, for example, as a consequence of conservation of energy. It is only in the special and general relativity where the concept of "negative energy" becomes problematic, when it ties up with the mass energy equivalence. Otherwise, negative energy means that, from conservation of energy, positive energy also exists in the system, because of the energy conservation law.
Thus the question has "unexpected" meaning only if it is asking whether negative masses exist, not in the formulation of your question, where the answer is "yes".
It is only negative masses that would behave the way you describe:
as it would move in the opposite direction from its momentum and accelerate in the opposite direction of an applied force. It would also warp space time in the opposite way from ordinary matter meaning that it would warp space time in such a way that it will produce repulsive Gravity.
All these are results from the existence of negative mass, not negative energy, which exists in all classical systems.
The problem of whether antiparticles have also negative gravitational mass is under experimental exploration.
We observed the times and positions at which 434 trapped antihydrogen atoms escaped our magnetic trap, and searched for the influence of a gravitational force. Based on our data, we can exclude the possibility that the gravitational mass of antihydrogen is more than 110 times its inertial mass, or that it falls upwards with a gravitational mass more than 65 times its inertial mass.
The experiments have a long way to go.
In what you read, Guth was developing the concept of inflation, and dark energy. These concepts were not fully understood nor developed when he first wrote, so he used some vague terms for them. Negative gravity is now called dark energy. In the development of physics since then, Dark Energy is accepted to have a positive energy density, therefore the answer to your question is a "yes".
Here is a link to a related question about energy conservation in GR, where the answers address your question very clearly: Energy conservation in General Relativity
Note both this link, and the other one provided by @user4552 in the comments, note that the "Zero Energy Hypothesis" for our universe, which Guth referred to, is untrue, as the energy of the overall universe in GR is undefinable, and energy is not universally conserved in GR.
Note, for both links, one Physics SE poster argued that energy IS conserved, but only referenced his own paper -- which is NOT the way one shows consensus of physics experts SUPPORTS a POV, but instead tends to show the consensus rejects the POV.
Best Answer
For one, negative mass would still be attracted to positive mass, but the positive mass would be repelled. This would lead to the negative mass "following" the positive mass.
Why is this? The force is a repulsive one. But we also have the fact that $\vec F=m\vec a$. Since one of the bodies has negative mass, it will be attracted.
If there was a body of negative mass less massive (talking about absolute values here) than our positive massed-universe, it would move much faster and eventually "catch up" with ours. If it had the same absolute value of mass, both would keep accelerating and it would never catch up. If it was larger, it would be left behind eventually.
The lack of any large quantities of negative mass in our universe excludes the case of a body of negative mass having caught up with us. The lack of any acceleration of the universe signifies that there isn't any large body of negative mass with absolute value less than or equal to the mass of the universe. So, if there is any large body of negative mass, it has larger mass than our universe and it is separated from our universe.
This is a part of his MAthematical Universe hypothesis. Note that it's just a hypothesis, it isn't backed by much concrete evidence yet.
This is true--it lets one create energy out of thin air by introducing a body with negative mass to the system, but this doesn't conflict the principle of conservation of energy as the body has it's own energy become more negative.
I'm not too sure about the energy conditions, I'm not strong in General Relativity. But the second half is just about some strangeness attached to it and some confusion regarding terminology (relating to what I explained near the top of this answer).
Finally, vacuum fluctuations (which exist) can have a net negative energy density, so these have "negative mass", in a way. But these aren't easy to harness with current technology.