It seems to depend on the reason for denying their illness...
... a term referring to behaviors, values, feelings that are in
harmony with or acceptable to the needs and goals of the ego, or
consistent with one's ideal self-image. ... Many personality disorders
are considered to be egosyntonic ... Anorexia ... is also considered
egosyntonic because many of its sufferers deny that they have a
People with personality disorders for example, may be fully aware and accepting of their behaviour, they simply don't see it as a problem.
Denial / abnegation:
... is used for a psychological defense mechanism ..., in which a
person is faced with a fact that is too uncomfortable to accept and
rejects it instead, insisting that it is not true despite what may be
overwhelming evidence. ... In denial, an individual does not see or is
mostly unconscious of existence of the truth or fact. The choice to
refuse reality, then, is unconscious as well.
Someone may deny their illness because it makes them uncomfortable in some way - eg, they are embarrassed about it or concerned about negative perceptions by others.
... a deficit of self-awareness, a condition in which a person who
suffers a certain disability seems unaware of the existence of his or
her disability. ... Anosognosia is relatively common following
different etiologies of brain injury ..., but can appear to occur in
conjunction with virtually any neurological impairment.
In Anton–Babinski syndrome (visual anosognosia) for example, a person is blind, but nonetheless claims to be able see, and this is due to brain damage in the visual cortex.
Hypochondria is undue alarm about any physical or psychological symptoms, no matter how minor they may be. So I would argue that the best counterpart is minimisation - a cognitive distortion:
... is the opposite of exaggeration. ... downplaying the significance
of an event or emotion ...
A medical condition that fits this description is anosodiaphoria:
... a condition in which a person who suffers disability due to brain
injury seems indifferent to the existence of their handicap. ... a
disorder of the body schema in which patients verbally acknowledge a
clinical problem ... but fail to be concerned about it.
So while a hypochondriac makes a big deal out of a small problem, anosodiaphoria is making a small deal out of a big problem.
Also see hyperchondria.