Depends on what you mean, but the short answer is "not really".
One study that seems to get peddled around as evidence that cold showers boost the immune system is this one, although the study authors don't state that. They state a slight or minimal effect in bloodwork, with an undetermined effect. From the abstract, emphasis added:
The aim of this study was to investigate whether or not the human immune system can be activated by a noninfectious stimulus, thereby improving the physiological status of the individual. The effect of a single cold water immersion (14 degrees C for 1 h) on the immune system of athletic young men, monitored immediately after immersion, was minimal. With the continuation of the cold water immersions (three times a week for a duration of 6 weeks) a small, but significant, increase in the proportions of monocytes, lymphocytes with expressed IL2 receptors (CD25) and in plasma tumour necrosis factor alpha content was induced. An increase in the plasma concentrations of some acute phase proteins, such as haptoglobin and haemopexin, was also observed. After 6 weeks of repeated immersions a trend towards an increase in the plasma concentrations of IL6 and the amount of total T lymphocytes (CD3), T helper cells (CD4), T suppressor cells (CD8), activated T and B lymphocytes (HLA-DR) and a decrease in the plasma concentration of alpha 1-antitrypsin was observed. Concentrations of IL1 beta, neopterin, C-reactive protein, orosomucoid, ceruloplasmin, macroglobulin, immunoglobulins (IgG, IgM, IgA) and C3, C4 components of the complement, as well as the total number of erythrocytes, leucocytes, granulocytes and neutrophils showed no significant changes after the repeated cold water immersions. It was concluded that the stress-inducing noninfectious stimuli, such as repeated cold water immersions, which increased metabolic rate due to shivering the elevated blood concentrations of catecholamines, activated the immune system to a slight extent. The biological significance of the changes observed remains to be elucidated.
The latest study on the matter I could find, which is much more on point for those of use who don't shower by being immersed in water for an hour, is here. It's also been recently cited in news articles, such as Cold Showers Lead to Fewer Sick Days at Harvard Business Review.
The Effect of Cold Showering on Health and Work: A Randomized Controlled Trial (emphasis added again):
The aim of this study was to determine the cumulative effect of a routine (hot-to-) cold shower on sickness, quality of life and work productivity.
Between January and March 2015, 3018 participants between 18 and 65 years without severe comorbidity and no routine experience of cold showering were randomized (1:1:1:1) to a (hot-to-) cold shower for 30, 60, 90 seconds or a control group during 30 consecutive days followed by 60 days of showering cold at their own discretion for the intervention groups. The primary outcome was illness days and related sickness absence from work. Secondary outcomes were quality of life, work productivity, anxiety, thermal sensation and adverse reactions.
79% of participants in the interventions groups completed the 30 consecutive days protocol. A negative binomial regression model showed a 29% reduction in sickness absence for (hot-to-) cold shower regimen compared to the control group (incident rate ratio: 0.71, P = 0.003). For illness days there was no significant group effect. No related serious advents events were reported.
A routine (hot-to-) cold shower resulted in a statistical reduction of self-reported sickness absence but not illness days in adults without severe comorbidity.
Twenty serious adverse events were reported, that were all considered unrelated to the intervention. One participant in the 90 seconds intervention group died unexpectedly of occult chronic pulmonary embolism at 56 days follow-up.
In this pragmatic randomized controlled trial, routinely showering (hot-to-) cold resulted in a 29% reduction of self-reported sick leave from work but not illness days at 90 days follow-up in adults without severe comorbidity. The contrast between the results of both primary outcome parameters is suggestive of the fact that the intensity rather than the duration of symptoms is modulated by the intervention. Regular physical activity resulted in a 35% reduction of sickness absence.
So, no one's found a statistically significant correlation between cold showers and preventing illness, but there might be some correlation in "modulating" the symptoms, a somewhat smaller correlation than regular physical activity. So, cold showers won't prevent a cold or the flu, but according to this study, they are correlated with a ~29% fewer sick days (presumably while infected).
The study also found no statistically significant correlation between cold showers and death by pulmonary embolism. Personally, given that cold showers correlate to a significantly insignificant reduction in illness, and a statistically insignificant chance of death, I'll be continuing my use of warm water in the mornings, but your mileage may vary.